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Theorem genpelv 9701
 Description: Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
genp.1 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
genp.2 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
Assertion
Ref Expression
genpelv ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑔,,𝐴   𝑥,𝐵,𝑦,𝑧,𝑔,   𝑥,𝑤,𝑣,𝐺,𝑦,𝑧,𝑔,   𝑔,𝐹   𝐶,𝑔,
Allowed substitution hints:   𝐴(𝑤,𝑣)   𝐵(𝑤,𝑣)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑣)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,)

Proof of Theorem genpelv
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 genp.1 . . . 4 𝐹 = (𝑤P, 𝑣P ↦ {𝑥 ∣ ∃𝑦𝑤𝑧𝑣 𝑥 = (𝑦𝐺𝑧)})
2 genp.2 . . . 4 ((𝑦Q𝑧Q) → (𝑦𝐺𝑧) ∈ Q)
31, 2genpv 9700 . . 3 ((𝐴P𝐵P) → (𝐴𝐹𝐵) = {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)})
43eleq2d 2673 . 2 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ 𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)}))
5 id 22 . . . . . 6 (𝐶 = (𝑔𝐺) → 𝐶 = (𝑔𝐺))
6 ovex 6577 . . . . . 6 (𝑔𝐺) ∈ V
75, 6syl6eqel 2696 . . . . 5 (𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
87rexlimivw 3011 . . . 4 (∃𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
98rexlimivw 3011 . . 3 (∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺) → 𝐶 ∈ V)
10 eqeq1 2614 . . . 4 (𝑓 = 𝐶 → (𝑓 = (𝑔𝐺) ↔ 𝐶 = (𝑔𝐺)))
11102rexbidv 3039 . . 3 (𝑓 = 𝐶 → (∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
129, 11elab3 3327 . 2 (𝐶 ∈ {𝑓 ∣ ∃𝑔𝐴𝐵 𝑓 = (𝑔𝐺)} ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺))
134, 12syl6bb 275 1 ((𝐴P𝐵P) → (𝐶 ∈ (𝐴𝐹𝐵) ↔ ∃𝑔𝐴𝐵 𝐶 = (𝑔𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173  (class class class)co 6549   ↦ cmpt2 6551  Qcnq 9553  Pcnp 9560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-ni 9573  df-nq 9613  df-np 9682 This theorem is referenced by:  genpprecl  9702  genpss  9705  genpnnp  9706  genpcd  9707  genpnmax  9708  genpass  9710  distrlem1pr  9726  distrlem5pr  9728  1idpr  9730  ltexprlem6  9742  reclem3pr  9750  reclem4pr  9751
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